Service Engineering January Laws of Congestion

Transcription

1 Service Engineering January Laws of Congestion The Law for (The) Causes of Operational Queues Scarce Resources Synchronization Gaps (in DS PERT Networks) Linear effects of scarcity and log effects of synchronization The Laws of Conservation Little s Law for Customers, Service providers and Managers: L = λ W E[S] Little s Law for the Offered Load (Utilization Profiles): ρ = λ N Laws of Completely Random Arrivals Levy/Watanabe Axioms of Randomness The Law of Poisson Counting (Law of Rare Events) The Law of Independent Memoryless (Exponential) Inter arrivals The Brownian Law of Rescaling & Centering High rate Arrivals The Law of Time Changing Time homogeneous Arrivals The Law of Accelerating Time inhomogenous Arrivals (or, Smoothing out Stochastic Variability around Predictable Variability) The Laws of Decomposition Superposition Laws of Sampling Random Sampling: Wolff s PASTA = Poisson Arrivals See Time Averages Biased Sampling: Costs of Randomness; (Coefficient of Variation; Form Factor) Laws of Human Service Durations What is Service Duration? The Theoretical Law of Phase Type Durations Empirical Laws of Exponential or Log Normal Service Durations The Law of Consistent Incentives: Abandoning Service providers Laws for Service Systems with Abandonment The Law of the Fittest survive (and Wait Less Much Less); The Linear Law of Abandonment rates for Casual/Uninformed Customers; Palm s Law of Irritation (Survival functions and Hazard rates); (The) Impatience/ Loyalty Index; The Law of Information shocks (or The Phases of Patience: Optimism, Facing Reality, Accepting Reality) (or The Phases of Patience: Customers Heterogeneity); The Adaptivity/Learning Cycle (Anticipation, Experience, Perception,...). 1

2 The Two moment Law for Average Congestion, in Efficiency Driven Systems Congestion Index (Efficiency vs. Quality, in the face of Stochastic Variability.) E[W a + Cs 2 q ] E[L q ] 1 ρ ρ C = E[S] N (1 ρ) 2 N Khintchine Pollaczek (Exact in M/G/1; ρ = P {W q > 0}, but only in numerator ) Allen Cunneen Approximation, for not too many E Driven Servers (GI/GI/N) C2 s 2 a + C 1 = E[S] P M/M/N{W q > 0} C 2 2 a + C E s GI/GI/N [W q ] E M/M/N [W q ] 2 (1 ρ) 2 N The Invariance Exponential Law for Long Delays Kingman s Exponential Law for the Distribution of Delay 80:20 Rules : Tails of The Delay Distribution in Efficiency Driven Operations The Law of Simplicity : Simple Theoretical Models describe Ideal Robust Realities. QED Q s (= Quality and Efficiency Driven Queues). 2 2

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9 Theoretical Congestion Curves: Staffing Tools (4CallCenters) Economies of Scale Average Waiting Time But Only of Those Who Wait E[W q W q > 0] (Load: 10 per server) 200 No. of Servers = Waiting Time In Seconds Arrivals Rate 12

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11 What is Service Time / Duration? Operations Time In a Hospital Operations Time Histogram: Frequency 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% 0% AVG: 2.08 Hours STD: 4.12 Hours Sample Size: 4347 CV >> Hours Operations Time - Morning vs. Afternoon: Queues Reduction Regular AM Hours 3 2 PM 1 0 EEG Orthopedics Surgery Blood Surgery Plastic Surgery Department Heart/Chest Neuro-Surgery Eyes E.I. Surgery Surgery Afternoon, by Case Morning, by Hour Ethical? Even Doctors Can Manage! 39

12 Forms Address Change What is Service Time? Bank Classification of Continued Calls Total: 2,400 calls 20% of all calls. Connecting Or Disconnecting Calls Listing Connecting Secretary Long Distance Calls Options Free Time Program Instructio ns Manual Monthly Invoice Means of Payment Call Type 27 Misc. Technical Problems # Calls

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15 Palm's Law of Irritation: I h (t) t R

16 Empirically-Based Theory Linear pattern observed: P{Abandon} = C E[Wait] Theory: Average Patience = 1/C in Erlang-A, else? 59

17 PATIENCE INDEX How to Define? Measure? Manage? Statistics Time Till Interpretation 360K served (80%) 2 min.? must = expect 90K abandon (20%) 1 min.? willing to wait Time willing to wait of served is censored by their wait. Uncensoring (simplified) 360K Willing to wait = = 9 min. 90K Expect to wait K 1 = = 2.25 min. 360K 4 Patience Index = time willing time expect definition = 4 = # served/wait > 0 # abandon/wait > 0 measure Supported by ongoing research (Wharton). 64

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19 Figure 24: Patience Indices: empirical vs. theoretical (R 2 =0.94) 8 Theoretical.Index Empirical.Index 57

20 Human behavior Delayed Abandons (IVR) Balking (New Customers) Learning (Internet Customers) 60

21 Customer-Focused Queueing Theory 200 abandonment in Direct-Banking Not scientific Reason to Abandon Actual Abandon Perceived Abandon Perception Time (sec) Time (sec) Ratio Fed up waiting (77%) Not urgent (10%) Forced to (4%) Something came up (6%) Expected call-back (3%) Rational Abandonment from Invisible Queues (with Shimkin). 21

22 Fitting a Simple Model to a Complex Reality 68